6. P is known from 5 and 3 — TheMadFool

How do you get "P is known" from "Assume proposition Q: P is unknown" and "Assume Q is known"?

Q is the statement ''P is unknown''...

So, if you know Q, you must know P

For example P: there is life on other worlds

Q: There is life on other worlds is unknown

So, knowing Q means you know P

To know that P is to know that P is true.

If P is "there is life on other worlds" and if Q is "the truth of P is unknown" then to know that Q is true is not to know that P is true. In fact, to know that Q is true is to not know that P is true.

Perhaps I didn't word it well. Let's try again:

P is a truth: The USA has 1 president

Q: P is unknown: It is unknown that The USA has 1 president.

Now assume Q is known: We know It is unknown that The USA has 1 president.

Now we know that P is true AND that it was unknown.

I interpret your axioms as follows:
1. x --> ◊K(x) (All truths are knowable ie. it is possible that somebody knows x at some time.) (axiom)
2. P (Proposition P is a truth) (axiom)
3.a) ~K(P) (P is unknown) (axiom)
3.b) Q:=~K(P) (definition of Q)
4. ◊K(Q) (Theorem from 1. and 3.)
5. K(Q) (Q is known) (axiom)

6. Theorem: K(P) (P is known)
Can you please clarify the proof of that theorem using this formal language? I don't think these axioms are enough to prove statement 6. Maybe your logic differs from this. In fact (6.) is not a theorem of the above system.

As long as I understand your premises are stronger than that of Fitch's paradox in existential terms.
While Fitch's paradox only assumes P and Q; you assume K(Q) also.

So first I don't see how we can prove 6. and even if we can get a contradiction from your axioms they may be stronger than the axioms of Fitch's paradox.

To be honest I don't really understand the paradox explained here:
https://plato.stanford.edu/entries/fitch-paradox/#ParKno
At (6) they get a contradiction and from that we can prove every statement (can't we?)
But they keep proving for 5 more steps for no apparent reason.

We know It is unknown that The USA has 1 president.

Now we know that P is true AND that it was unknown. — TheMadFool

That doesn't follow. If we know that we don't know that the USA has 1 president then we don't know that the USA has 1 president, and so we don't know that P is true.

What is driving

At (6) they get a contradiction and from that we can prove every statement (can't we?)
But they keep proving for 5 more steps for no apparent reason. — Meta

There is a good reason actually. It's because (6) is a contradiction that the premise of the argument must be discharged: that is, negated. That's how reductio proofs work. Step (7) just is the negation of the premise that logically led to the contradiction (assuming the validity of all the deductive principles that had been used in the previous steps).

(On edit: this was of course a comment regarding the proof as presented in the SEP article linked in Meta's post.)

There's an extra step you're missing.

That P is unknown, is known (Q). P being a part of Q, is, let's say, automatically known.

For example:

P: elephants are mammals

Q: P is unknown i.e it is unknown that elephants are mammals.

Then, if you know Q: it is known that it is unknown that elephants are mammals. That means we now know that elephants are mammals

Then, if you know Q: it is known that it is unknown that elephants are mammals. That means we now know that elephants are mammals — TheMadFool

No it doesn't.

Do you know what my hair colour is? No. Do you know that you don't know what my hair colour is? Yes. Therefore you know what my hair colour is? No.

If you know that you don't know X then you don't know X.

The goal of the argument was to show that {(KP),(NonO),(A),(B),(C),(D)} is inconsistent. To show that they looked for a contradiction. At step (6) they got a contradiction so for me that is the end of the proof. Their proof didn't end there and they found a contradiction at (9). This was weird for me; however the problem is not a big deal just a small technical detail.

Basically they prove {X,Y}⊢⊥ by proving {X,Y}⊢⊥=>~X by reductio; and then {X,~X}⊢⊥
I think it wasn't the most elegant method but as I said it is not a big deal.

Allow me to rephrase your example:

P: the color of your hair is red

Q: P is unknown: It is unknown that the color of your hair is red

Assume you know Q: It is known that it is unknown that the color of your hair is red.

By knowing Q, you now know the nested truth, which is: the color of your hair is red.

I think your mistake is not forming a proposition. Propositions like P have to be declarative sentences and not questions, as in your example i.e. ''what is the color of my hair?'' is a question

I think your grammar is confusing you (consider the apparent difference between "I don't know that aliens exist" and "I don't know if aliens exist").

The below formulation should make it clearer.

P: the colour of my hair is red
Q: the truth-value of P is unknown (the truth-value of "the colour of my hair is red" is unknown).

Assume you know Q: it is known that the truth-value of "the colour of my hair is red" is unknown.

By knowing Q, you don't know the truth-value of "the colour of my hair is red" (and so, if it's true, don't know that it's true).

Your formulation seems to conflate a meta-description (where we assert that P is true, and that some unspecified person doesn't know that P is true) with an ordinary description (where I assert that I don't know if P is true).

I think your mistake is not forming a proposition.

I think your mistake is in forming a proposition. It creates a grammatical confusion that can result in the contradiction given in 7). My wording here is far simpler, and shows that knowing that something is unknown doesn't entail knowing the unknown.

I think the problem is with these two premises:

2. Proposition P is a truth

3. Assume proposition Q: P is unknown

If we take P to be "my hair is red" then we have:

2. "my hair is red" is true

3. Assume proposition Q: "my hair is red" is unknown

The problem is that 3 is ambiguous. Does it mean "the truth-value of 'my hair is red' is unknown" or does it mean "'my hair is red' is an unknown truth"? Note that the latter interpretation doesn't follow from the second premise. What you seem to have done is conflated 2 and P.

To better present Fitch's paradox, premise 3 should read "Assume proposition Q: P is an unknown truth".

To better present Fitch's paradox, premise 3 should read "Assume proposition Q: P is an unknown truth" — Michael

Amounts to the same thing. Asserting P is the same as asserting P is true.

So, when I say Q: P is unknown, I mean P is an unknown truth. To show that that's exactly what I mean ''P is an unknown truth'' means ''P is a truth AND P and P is unknown''. There's no need to say ''P is a truth''. That's redundant.

Anyway, now that you agree there's a paradox, how do we solve it?

There's no need to say ''P is a truth''. That's redundant. — TheMadFool

Then "P is false" would be a contradiction. So for any proposition P, P must be true. But that doesn't work. We're quite correct in saying "P is false", where P is the proposition that London is the capital city of France. You're conflating use and mention.

Amounts to the same thing. Asserting P is the same as asserting P is true.

This isn't a case of asserting P, though. This is a case of saying something about P – it's mention, not use.

I think there is nothing paradoxical in the possibility that there are true sentences which are not knowable. The absolute conistency of arithmetic or set theory could be examples for this. The absolute consistency of these theories can be true but since we can only prove relative consistency we will never know if they are true or not.
edit: Another example: Let's say there are x particles in the universe. There are formulas which's length is 10^x^x^x^x^x. Some of them may be true. I don't think we would ever know the truth value of each of them.

From what I know:

Everyday experience evidences that we don't make statements like '' It is true that the sun is shining''. Rather we say ''the sun is shining''.

In formal logic too, an assertion doesn't have to qualify itself with the truth value ''true'' i.e. the proposition P means P is true.

However, falsity needs to be made explicit because simply stating P: ''the sun is shining'' is taken as a truth claim . Thus we say ''it is false that the sun is shining''. In logic this is ~P.

No contradiction.

Then "P is false" would be a contradiction — Michael

''P is false'' is simply saying ''P is true is false''. So, P is false. There's no contradiction.

Again, you're conflating use and mention. When we say "I don't know P" we're saying something about the proposition P, not asserting it. We're not saying "it is raining, but I don't know it"; we're saying "I don't know if it is raining".

Fitch's paradox is about unknown truths, not unknown statements. He wants to show that, if all truths are knowable, then all truths are known.

Here's the correct formulation of the paradox based on the Stanford article cited by Meta.

Fitch's argument proves that if one assumes that all truths are knowable in principle, then it follows that they must be knowable in actuality (that is, everyone is omniscient) - which is of course an absurdity. Here's how it goes.

First we assume the knowability principle:

KP. For every proposition P, it is possible to know P.

Now let us assume that the following conjunction is true (which by itself is a possible state of affairs):

a. P is true.
b. No one knows that P is true.

Now, according to the knowability principle, it is possible to know every P, so it follows that it is possible to know the conjunction of a. and b.. But it is impossible to know this conjunction: you cannot know that P and also know that nobody knows that P (since you do know it), so it follows that the conjunction of a. and b. is unknowable. But the conjunction says that there is a truth that no one knows, and this seems right, but the argument shows that it is impossible. So it follows that either we are omniscient or that the knowability principle is false.

And the paradox mainly consist in the fact that the knowability principle shouldn't entail such an absurd conclusion by a simple deductive argument (or so it seems - even if one doesn't accept the knowability principle, it still seems strange that it should entail such a conclusion).

I don't understand why did they choose that single sentence to show the absurdity of the argument. Since EVERY formula is provable in a contradictory logical system (at least in first order logic- but modal logic cant be much different) 1+1=2 is also provable which is absurd for every rational human being- and even for theists. (Since theists believe in an omniscient being omniscience is natural for them.)

The paradox arises from the fact that the logical language being used is unconstrained second order, and IIRC second-order logical languages are inconsistent unless subjected to some constraints.

That the language is second order can be seen from the implicit predicate in the statement

KP. For every proposition P, it is possible to know P. — Fafner

This requires the existence of a unary predicate Know, which can take as argument any well-formed sentence in the language.

The paradox tells us about the problems of using unconstrained second-order languages, rather than telling us anything meaningful about knowledge.

If one of the standard ways of constraining higher-order logic to retain consistency (eg Russell's theory of Types), is invoked, the paradox disappears because some of the statements in the attempted proof cannot be made - they are syntax errors.

Here's my way of seeing it.

P: The statue of liberty exists: Statue of liberty exists is a truth

Q: P is an unknown truth: The statue of liberty exists is an unknown truth: The statue of liberty exists is a truth AND The statue of liberty exists is unknown.

Now the tricky part. I'll clarify as much as I can.

If I say ''Michael is a good man'', I mean to say, "Michael is a man'' AND ''Michael is good''. Decomposing compund statements into its parts is logically legitimate.

Consider now the statement: Q is known.

Q is known = it is known that P is an unknow truth = it is known that P is a truth AND P is unknown.

So, knowing Q, P is known to be true.

The paradox tells us about the problems of using unconstrained second-order languages, rather than telling us anything meaningful about knowledge — andrewk

So the problemis with logic? Why is it then that it's relatively easy to understand the paradox using simple logic as I have?

(Y)

Your logic seems to be accepted and understood by a single person in the world therefore it is not a classical one.

If one of the standard ways of constraining higher-order logic to retain consistency (eg Russell's theory of Types), is invoked, the paradox disappears because some of the statements in the attempted proof cannot be made - they are syntax errors. — andrewk

But there's no such thing as a theory of types, and there could be no "syntax errors" in a language (because every sentence in language can be potentially made sense of with the right interpretation).

It seems to me like a cheap trick to get out of a paradox by stipulating that you simply cannot say certain sentences. If you cannot put 'know' in front of every sentence then there should be a principled explanation why (because KP seems like a perfectly consistent thing to say, and some people even think that it is true), and so I do think that the paradox can teach us something interesting about knowledge.

Your logic seems to be accepted and understood by a single person in the world therefore it is not a classical one. — Meta

Perhaps there are errors in my version of the argument. I'm only familiar with sentential logic and a little bit of predicate logic. May be you can do better. I'd really like to see that. Thanks

In my first post I formalized your argument and turned out the deduction is incorrect.
Maybe you got the right intuition but we need a logic to create a logical frame for your intuition.

If Q means (P and ~K(P)) ((Q means P is true and not knowable))
And we assume K(Q)
Then ~K(P) immediately follows.
That is a contradiction indeed; the similar to the Fitch paradox but not the same.

Assume you know Q: It is known that it is unknown that the color of your hair is red.

By knowing Q, you now know the nested truth, which is: the color of your hair is red. — TheMadFool

It's amazing that after so many explanations you can't get this.

By knowing Q you know that is is UNKNOWN whether the color of someone's hair is red. That doesn't imply that you know that P is true. It implies just the opposite.